Understanding Parabola and Its Elements

A parabola is a set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The parabola is one of the four types of conic sections, the others being the ellipse, the circle, and the hyperbola.

Key Elements of a Parabola

Before diving into problems, let's understand the key elements associated with a parabola:

• Focus (F): A fixed point inside the parabola.
• Directrix: A fixed line outside the parabola.
• Vertex (V): The point where the parabola changes direction, located midway between the focus and the directrix.
• Axis of Symmetry: The line that passes through the focus and vertex, and divides the parabola into two symmetrical halves.
• Latus Rectum: A line segment perpendicular to the axis of symmetry, passing through the focus, and whose endpoints lie on the parabola.

Standard Forms of Parabola

There are two standard forms of a parabola, depending on its orientation:

1. Vertical Parabola: Opens upwards or downwards.
   • Equation: (y - k = a(x - h)^2), where ( (h, k) ) is the vertex.
2. Horizontal Parabola: Opens to the right or left.
   • Equation: (x - h = a(y - k)^2), where ( (h, k) ) is the vertex.

Formulas for Elements of a Parabola

The following table summarizes the formulas for finding the focus, directrix, vertex, and latus rectum of a parabola given its standard equation:

Element	Vertical Parabola (y = ax^2 + bx + c)	Horizontal Parabola (x = ay^2 + by + c)
Vertex (V)	((-b/2a, c - (b^2 - 4ac)/4a))	((c - (b^2 - 4ac)/4a, -b/2a))
Focus (F)	((-b/2a, c - (b^2 - 4ac)/4a + 1/4a))	((c - (b^2 - 4ac)/4a + 1/4a, -b/2a))
Directrix	(y = c - (b^2 - 4ac)/4a - 1/4a)	(x = c - (b^2 - 4ac)/4a - 1/4a)
Axis of Symmetry	(x = -b/2a)	(y = -b/2a)
Latus Rectum	Length: (	4a

Examples

Example 1: Vertical Parabola

Given the equation of a parabola (y = 2x^2 - 4x + 3), find the focus, directrix, vertex, and latus rectum.

Solution:

1. Vertex (V): Using the formula for the vertex, we have: [ h = -\frac{b}{2a} = -\frac{-4}{2 \cdot 2} = 1, \quad k = c - \frac{b^2 - 4ac}{4a} = 3 - \frac{(-4)^2 - 4 \cdot 2 \cdot 3}{4 \cdot 2} = 1 ] So, the vertex is (V(1, 1)).

2. Focus (F): The focus is given by: [ F\left(1, 1 + \frac{1}{4a}\right) = F\left(1, 1 + \frac{1}{4 \cdot 2}\right) = F\left(1, \frac{5}{4}\right) ]

3. Directrix: The directrix is a line parallel to the x-axis, given by: [ y = k - \frac{1}{4a} = 1 - \frac{1}{4 \cdot 2} = \frac{3}{4} ]

4. Latus Rectum: The length of the latus rectum is (|4a| = |4 \cdot 2| = 8).

Example 2: Horizontal Parabola

Given the equation of a parabola (x = -\frac{1}{2}y^2 + 3y - 2), find the focus, directrix, vertex, and latus rectum.

Solution:

1. Vertex (V): Using the formula for the vertex, we have: [ k = -\frac{b}{2a} = -\frac{3}{2 \cdot (-1/2)} = 3, \quad h = c - \frac{b^2 - 4ac}{4a} = -2 - \frac{3^2 - 4 \cdot (-1/2) \cdot (-2)}{4 \cdot (-1/2)} = -5 ] So, the vertex is (V(-5, 3)).

2. Focus (F): The focus is given by: [ F\left(-5 + \frac{1}{4a}, 3\right) = F\left(-5 - \frac{1}{4 \cdot (-1/2)}, 3\right) = F\left(-4, 3\right) ]

3. Directrix: The directrix is a line parallel to the y-axis, given by: [ x = h - \frac{1}{4a} = -5 - \frac{1}{4 \cdot (-1/2)} = -4 ]

4. Latus Rectum: The length of the latus rectum is (|4a| = |4 \cdot (-1/2)| = 2).

Practice Problems

1. Find the focus, directrix, vertex, and latus rectum of the parabola (y = 4x^2).
2. Determine the elements of the parabola given by the equation (x = -3y^2 + 6y + 7).
3. For the parabola with the equation (y^2 = 16x), calculate the coordinates of the focus and the equation of the directrix.

By understanding these formulas and practicing problems, one can master the topic of parabolas and their elements, which is essential for exams and further studies in mathematics.